Both health workers treated Liberian man Thomas Duncan, who died last week after becoming the first person diagnosed with Ebola in the US.
Meanwhile, the UN's Ebola mission chief says the world is falling behind in the race to contain the virus.
The World Health Organization (WHO) says 4,447 people have died from the outbreak, mainly in West Africa.
Sierra Leone, Liberia and Guinea have been hardest hit by the outbreak, which began in December 2013 but was confirmed in March.
Anthony Banbury told a special session of the UN Security Council on Tuesday that if Ebola was not stopped now, the world would "face an entirely unprecedented situation for which we do not have a plan".
President Barack Obama is due to hold a video conference with British, French, German and Italian leaders to discuss the Ebola crisis later on Wednesday.
In other developments:
New measures
The identity of the second Texas health worker has not yet been revealed.
Last weekend, officials announced nurse Nina Pham, 26, had been exposed to Ebola at a Dallas hospital when she treated Mr Duncan.
Doctors at the Health Presbyterian hospital said she was in good condition on Tuesday.
The health worker was immediately isolated after reporting a fever on Tuesday and people he had been in contact with will be monitored, the Texas State Department for Health said in a statement.
By James Gallagher, BBC health editor
This is a reminder that Ebola is a dangerous threat even within some of the best hospitals in world.
This is the second case of transmission in Texas, on top of the case in Spain.
Protective gear should minimise the risk to health workers treating Ebola patients.
But worryingly there have been reports of staff not getting the correct equipment or the necessary training.
These isolated cases should not become an outbreak as authorities in both countries are monitoring those who came into contact with infected people.
But each new case of transmission is asking the question - is enough being done to protect healthcare workers?
US officials previously said they were monitoring 48 contacts of the Liberian national and the healthcare workers who treated him.
The US Centers for Disease Control (CDC) expressed concern over the latest development in a statement, but added that it was "not unexpected that there would be additional exposures".
It has announced new measures to improve hospital preparedness for Ebola treatment, including an immediate response team that will travel to the site of any future Ebola diagnoses to hit the ground "within hours".
Nurses at the Dallas hospital say they worked for days without adequate protective clothing and received little guidance on how to prevent the spread of the virus.
Ebola patients treated outside West Africa*
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*In all but three cases the patient was infected with Ebola while in West Africa. Infection outside Africa has been restricted to health workers in Madrid and in Dallas. DR Congo has also reported a separate outbreak of an unrelated strain of Ebola.
Ebola basics: What you need to know
It comes after CDC chief Thomas Frieden said there had been a breach of protocol by health workers that led to the nurse becoming infected.
"The CDC is saying that protocols were breached, but the nurses are saying there were no protocols," the head of the national nurses union, Roseann DeMoro, told reporters on Tuesday.
Mr Frieden later said he regretted not sending a larger team of experts to Texas when the first case of Ebola was diagnosed.
There have been 8,914 Ebola cases overall, including the fatal cases, and the WHO says it expects this number to top 9,000 by the end of the week.
The WHO estimates its figures by taking the numbers of confirmed cases and multiplying them - from Guinea by 1.5, from Sierra Leone by 2 and from Liberia by 2.5 - to account for under-reporting.